Question:
<p>If 2x - 4y = 9 and 6x -12y + 7 = 0 are the tangents of same circle, then its radius will be</p>
<p style="display:inline"><span class="math-tex">\(\frac{17}{6 \sqrt{5}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{17}{3 \sqrt{5}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\sqrt{3}}{5}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2 \sqrt{5}}{3}\)</span></p>
Step-by-Step Solution
Key Concept: The perpendicular distance between two parallel tangents to a circle is equal to the diameter of the circle.
<p>We have,<br />
2x - 4y = 9 and 6x - 12y = -7<br />
<span class="math-tex">$\Leftrightarrow$</span> 2x - 4y <span class="math-tex">$=\frac{-7}{3}$</span><br />
Distance between two parallel lines<br />
<span class="math-tex">$=\left|\frac{c_{1}-c_{2}}{\sqrt{a^{2}+b^{2}}}\right|$</span><br />
<span class="math-tex">$=\left|\frac{9-\left(\frac{-7}{3}\right)}{\sqrt{(2)^{2}+(-4)^{2}}}\right|=\left|\frac{9+\frac{7}{3}}{\sqrt{4+16}}\right|=\left|\frac{\frac{34}{3}}{\sqrt{20}}\right|$</span> <span class="math-tex">$=\left|\frac{34}{\sqrt{180}}\right|$</span><br />
Since the tangents are parallel, therefore the distance between these two tangents will be their diameter<br />
<span class="math-tex">$\Rightarrow$</span> Diameter <span class="math-tex">$=\frac{34}{\sqrt{180}}=\frac{17}{3 \sqrt{5}}$</span><br />
<span class="math-tex">$\Rightarrow$</span> Radius <span class="math-tex">$=\frac{17}{6 \sqrt{5}}$</span></p>
Correct Answer: A